Gauss%E2%80%93Chebyshev%20formula

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11: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

§35.8(iii) ${{}_{3}F_{2}}$ Case

►

Kummer Transformation

… ►

Pfaff–Saalschütz Formula

… ►

Thomae Transformation

… ►Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions

{{}_{p}F_{q}}

and

{{}_{p+1}F_{p}}

of matrix argument. …

12: 15.9 Relations to Other Functions

… ►

Chebyshev

►

15.9.5

T_{n}\left(x\right)=F\left({-n,n\atop\frac{1}{2}};\frac{1-x}{2}\right).

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $x$ : real variable and $n$ : integer
Permalink:: http://dlmf.nist.gov/15.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

►

15.9.6

U_{n}\left(x\right)=(n+1)F\left({-n,n+2\atop\frac{3}{2}};\frac{1-x}{2}\right).

ⓘ

Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $x$ : real variable and $n$ : integer
Permalink:: http://dlmf.nist.gov/15.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

… ►

15.9.7

P_{n}\left(x\right)=F\left({-n,n+1\atop 1};\frac{1-x}{2}\right).

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $x$ : real variable and $n$ : integer
Permalink:: http://dlmf.nist.gov/15.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

… ►The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers. …

13: 13.31 Approximations

… ►

§13.31(i) Chebyshev-Series Expansions

►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of

M\left(a,b,x\right)

and

U\left(a,b,x\right)

that include the intervals

0\leq x\leq\alpha

and

\alpha\leq x<\infty

, respectively, where

\alpha

is an arbitrary positive constant. … ►

13.31.1

A_{n}(z)=\sum_{s=0}^{n}\frac{{\left(-n\right)_{s}}{\left(n+1\right)_{s}}{\left% (a\right)_{s}}{\left(b\right)_{s}}}{{\left(a+1\right)_{s}}{\left(b+1\right)_{s% }}(n!)^{2}}\*{{}_{3}F_{3}}\left({-n+s,n+1+s,1\atop 1+s,a+1+s,b+1+s};-z\right),

ⓘ

Defines:: $A_{n}(z)$ : expansion (locally)
Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.31.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.31(iii), §13.31 and Ch.13

… ►

13.31.2

B_{n}(z)={{}_{2}F_{2}}\left({-n,n+1\atop a+1,b+1};-z\right).

ⓘ

Defines:: $B_{n}(z)$ : expansion (locally)
Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.31.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §13.31(iii), §13.31 and Ch.13

…

14: 31.7 Relations to Other Functions

… ►

§31.7(i) Reductions to the Gauss Hypergeometric Function

… ►Other reductions of

\mathit{H\!\ell}

to a

{{}_{2}F_{1}}

, with at least one free parameter, exist iff the pair

(a,p)

takes one of a finite number of values, where

q=\alpha\beta p

. … ►

31.7.2

\mathit{H\!\ell}\left(2,\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-2\gamma+1% ;z\right)={{}_{2}F_{1}}\left(\tfrac{1}{2}\alpha,\tfrac{1}{2}\beta;\gamma;1-(1-% z)^{2}\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

►

31.7.3

\mathit{H\!\ell}\left(4,\alpha\beta;\alpha,\beta,\tfrac{1}{2},\tfrac{2}{3}(% \alpha+\beta);z\right)={{}_{2}F_{1}}\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}\beta% ;\tfrac{1}{2};1-(1-z)^{2}(1-\tfrac{1}{4}z)\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

… ►Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities

\zeta=K

K+i{K^{\prime}}

, and

i{K^{\prime}}

, where

K

and

{K^{\prime}}

are related to

k

as in §19.2(ii).

15: 20.11 Generalizations and Analogs

… ►

§20.11(i) Gauss Sum

►For relatively prime integers

m, n

with

n>0

and

m n

even, the Gauss sum

G(m,n)

is defined by … … ► … ►Similar identities can be constructed for

{{}_{2}F_{1}}\left(\tfrac{1}{3},\tfrac{2}{3};1;k^{2}\right)

{{}_{2}F_{1}}\left(\tfrac{1}{4},\tfrac{3}{4};1;k^{2}\right)

, and

{{}_{2}F_{1}}\left(\tfrac{1}{6},\tfrac{5}{6};1;k^{2}\right)

. …

16: 16.8 Differential Equations

… ►the function

w={{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

satisfies the differential equation … ►

w_{0}(z)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right),

… ►We have the connection formula … ►Analytical continuation formulas for

{{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

near

z=1

are given in Bühring (1987b) for the case

q=2

, and in Bühring (1992) for the general case. … ►

16.8.10

\lim_{|\alpha|\to\infty}{{}_{p+1}F_{q}}\left({a_{1},\dots,a_{p},\alpha\atop b_% {1},\dots,b_{q}};\frac{z}{\alpha}\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}% \atop b_{1},\dots,b_{q}};z\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.8.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §16.8(iii), §16.8 and Ch.16

…

17: 18.20 Hahn Class: Explicit Representations

… ►

§18.20(i) Rodrigues Formulas

… ►

18.20.5

Q_{n}\left(x;\alpha,\beta,N\right)={{}_{3}F_{2}}\left({-n,n+\alpha+\beta+1,-x% \atop\alpha+1,-N};1\right),

n=0,1,\dots,N

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.20(ii), §18.21(ii), §18.23, §18.26(ii), §18.27(ii), §18.38(iii)
Permalink:: http://dlmf.nist.gov/18.20.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.20(ii), §18.20 and Ch.18

►

18.20.6

K_{n}\left(x;p,N\right)={{}_{2}F_{1}}\left({-n,-x\atop-N};p^{-1}\right),

n=0,1,\dots,N

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.19, §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.20.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §18.20(ii), §18.20 and Ch.18

►

18.20.7

M_{n}\left(x;\beta,c\right)={{}_{2}F_{1}}\left({-n,-x\atop\beta};1-c^{-1}% \right).

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.21(ii), §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.20.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §18.20(ii), §18.20 and Ch.18

►

18.20.8

C_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,-x\atop-};-a^{-1}\right).

ⓘ

Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §13.6(v), §18.19, §18.20(ii), §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.20.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §18.20(ii), §18.20 and Ch.18

…

18: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

Gauss Formula

… ►

Reflection Formula

… ►Subject to the conditions (a)–(c), the function

f(\mathbf{T})={{}_{2}F_{1}}\left(a,b;c;\mathbf{T}\right)

is the unique solution of each partial differential equation … ►Systems of partial differential equations for the

{{}_{0}F_{1}}

(defined in §35.8) and

{{}_{1}F_{1}}

functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). … ►

35.7.10

{{}_{2}F_{1}}\left({\alpha a,\alpha b\atop\alpha c};\mathbf{T}\right)

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $c$ : complex variable and $\alpha\to\infty$ : parameter
Permalink:: http://dlmf.nist.gov/35.7.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(iv), §35.7 and Ch.35

…

19: 16.16 Transformations of Variables

… ►

§16.16(i) Reduction Formulas

►

16.16.1

{F_{1}}\left(\alpha;\beta,\beta^{\prime};\beta+\beta^{\prime};x,y\right)=(1-y)% ^{-\alpha}{{}_{2}F_{1}}\left({\alpha,\beta\atop\beta+\beta^{\prime}};\frac{x-y% }{1-y}\right),

ⓘ

Symbols:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function and ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function
Permalink:: http://dlmf.nist.gov/16.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.16(i), §16.16 and Ch.16

►

16.16.2

{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\beta^{\prime};x,y\right)=(1-y% )^{-\alpha}{{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};\frac{x}{1-y}\right),

ⓘ

Symbols:: ${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$ : second Appell function and ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function
Permalink:: http://dlmf.nist.gov/16.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.16(i), §16.16 and Ch.16

… ►

16.16.5

{F_{3}}\left(\alpha,\gamma-\alpha;\beta,\gamma-\beta;\gamma;x,y\right)=(1-y)^{% \alpha+\beta-\gamma}{{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};x+y-xy\right),

ⓘ

Symbols:: ${F_{3}}\left(\NVar{\alpha},\NVar{\alpha^{\prime}};\NVar{\beta},\NVar{\beta^{% \prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)$ : third Appell function and ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function
Permalink:: http://dlmf.nist.gov/16.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §16.16(i), §16.16 and Ch.16

… ►See Erdélyi et al. (1953a, §5.10) for these and further reduction formulas. …

20: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►Gauss and Legendre conjectured that

\pi\left(x\right)

is asymptotic to

x/\ln x

x\to\infty

: …(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►

Table 27.2.1: Primes.

► ►►

$n$	$p_{n}$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
…

► ►

Table 27.2.2: Functions related to division.

► ►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

►